From a string theory perspective, the Riemannian metric \(g_{\mu \nu }\)  is only a fraction of the massless sector of closed strings which should further contain a skew-symmetric \(B\) -field and a scalar dilaton \(\phi \) . The theory implies a crucial symmetry, \(\mathbf {O}(D,D)\)  with spacetime dimension \(D\) , which transforms the trio of \(\lbrace g,B,\phi \rbrace \)  into one another [1]}, [2]}. It calls for an \(\mathbf {O}(D,D)\)  singlet, \(e^{-2d}=\sqrt{-g}e^{-2\phi }\) , and an \(\mathbf {O}(D,D)\)  tensor,
\({\cal H}_{AB} = \scalebox {0.9}{\!\left(\!\begin{array}{cc}g^{-1} & - g^{-1}B \\B g^{-1} & g - B g^{-1} B\end{array}\!\right)\!=\! \left(\!\!\begin{array}{cc}1 & 0 \\B &1\end{array}\!\right) \!\!\left(\!\!\begin{array}{cc}g^{-1} &\!0 \\0 &\!g\end{array}\!\!\right) \!\!\left(\!\!\begin{array}{cc}1 & -B\\0 &1\end{array}\!\!\right)}\,,\) 
